Proof by Contradiction
Concept Explained: Proof by Contradiction

Proof by Contradiction, also known as Reductio ad absurdum, is a method of demonstrating that a statement is true by showing that it would lead to a contradiction if it were false.

This form of proof is useful when it is simpler to assume the opposite of the statement and then arrive at an impossible or absurd consequence.
Using Proof by Contradiction

To use Proof by Contradiction, assume that the proposition you wish to prove is false.

This assumption is added to your existing set of known truths and you then explore the logical consequences.

You are searching for a logical contradiction, which can be defined as a result or statement that conflicts with a previously established fact.

Finding this contradiction means that the assumption (that the proposition is false) must be wrong.

Therefore, if assuming the statement is false leads to a contradiction, the original statement must be true.
Example of Proof by Contradiction

As a direct illustration, let’s prove that there is no smallest positive rational number.

Assume towards contradiction, that there is a smallest positive rational number. Let’s call it ‘a’.

Then ‘a/2’ must be smaller than ‘a’, which contradicts the assumption that ‘a’ is the smallest positive rational number.

So, our assumption is contradicted and we conclude that there cannot be a smallest positive rational number.
Impact of Proof by Contradiction

A Proof by Contradiction might not give you a direct construction or an explanatory reason why something is the case.

This kind of proof can be counterintuitive, as you assume the opposite of what you want to prove.

Remember that Proof by Contradiction is widely accepted and used in all branches of mathematics as a powerful proof technique.
Tips for Constructing a Proof by Contradiction

Keep a clear path of logic and don’t lose sight of what you’re trying to prove. It’s easy to get lost while exploring the consequence of the assumption.

Be open to finding contradictions in unexpected places. Your contradiction could involve concepts or facts outside the immediate proposition.

Be precise and clear in identifying and stating your contradiction. Proofs rely on solid logical steps, so ensure your contradiction is unequivocally contradictory.

Don’t forget to conclude your proof by stating the original proposition has been proven true, because the assumption that it was false led to a contradiction.